Multiple quantum products in toric varieties

We generalize the author's formula for Gromov-Witten invariants of symplectic toric manifolds (see math.AG/0006156) to those needed to compute the quantum product of more than two classes directly, i.e. involving the pull-back of the Poincar\'e dual of the point class in the Deligne-Mumford spaces $ \mathcal{M}_{0,m} $.


Introduction
Let (X, ω) be a symplectic manifold with compatible almost-complex structure J. If g and m are non-negative integers, we denote by M g,m the Deligne-Mumford space of genus-g curves with m marked points. If furthermore A ∈ H 2 (X, Z) denotes a degree-2 homology class of X, M g,m (X, A) will be the space of stable genus-g J-holomorphic maps to X with homology class A. The Gromov-Witten invariants of X (see for example [RT95], [Beh97]) are multi-linear maps Here the integration on the right hand side is not over the entire moduli space but over the so-called virtual fundamental class.
For the case of (X, J) being a smooth projective variety with a (C * )-action, Graber and Pandharipande have proven (see [GP99]) that Bott-style localization techniques apply to the integral in (1). Their techniques can easily be extended to torus actions, so in particular they apply to smooth projective toric varieties (for toric varieties see for example [Ful93], [Oda88]).
In [Spi99a] (also see [Spi99b]), using these localisation techniques for the virtual fundamental class, we have proven an explicit combinatorial formula of the genus-0 Gromov-Witten invariants for smooth projective toric varieties for the cases when β = 1 ∈ H 0 (M 0,m , Q). In this note we will derive a similar formula for the case where the class β is the maximal product of (the Chern class of) cotangent lines to the marked points, that is for those classes β which are Poincaré dual to a finite number of points in M 0,m .
Knowing the Gromov-Witten invariants for β = P. D.(pt) makes computations of products in the (small) quantum cohomology ring easier. For two cohomology classes γ 1 , γ 2 ∈ H * (X, Q), their quantum product is defined to be where the inner sum runs over a basis (δ i ) of H * (X, Q). Here (δ ∨ i ) denotes the basis of H * (X, Q) dual to (δ i ). It is easy to show that the product of more than two classes is given by where P. D.(pt) denotes the Poincaré dual of a point in M 0,r+1 . By Witten's conjecture (see [Wit91], proven by Kontsevich in [Kon92]), we know that (a multiple of) the class P. D.(pt) in H * (M 0,r+1 ) can be expressed as the product of cotangent line classes, hence the invariants in (2) can be computed directly by the formula proposed in this note.
The techniques used in this note also yield the invariants for β being a different product of Chern classes of such cotangent line bundles. However, it seems to be much more to difficult to formalize such a more general approach. For the sake of a (hopefully) better exposition of the key ideas, we leave the more general case to the interested reader.
The structure of the paper is as follows: In Section 2 we recall some results on toric varieties, mostly to fix our notation. In Section 3 we quickly describe the fixed point components of M 0,m (X Σ , A) with respect to the action induced from X Σ . In Section 4 we recall the localization results for toric varieties, to apply them in Section 5 to the case where β = P. D.(pt). In Section 6 we finally give the formula for the Gromov-Witten invariants for symplectic toric manifolds in this case, and in Section 7 we illustrate the formula on the example of P P 1 (O ⊕(r−2) ⊕ O(1) ⊕ O(1)); as an interesting byproduct, we derive the quantum cohomology ring of this variety (also using recent results of [CMR01]), which surprisingly coincides with Batyrev's ring stated in [Bat93].

Preliminaries of toric varieties
We will quickly recall some facts about toric varieties and mostly introduce our notation -our standard references for this section are [Bat93], [Ful93] and [Oda88].
Let X Σ be a smooth projective toric variety of complex dimension d, given by the fan Σ. Choose a class ω in the Kähler cone of X Σ , and let ∆ ω be the corresponding moment polytope. On the variety X Σ , the d-dimensional torus T N := (C * ) d acts effectively, and the (irreducible) subvarieties of X Σ that are left invariant under this action are in one-to-one correspondence with the facets of the polytope ∆ ω . Moreover, the T N -invariant divisors (which are in one-to-one correspondence to faces of ∆ ω ) generate the cohomology ring H * (X Σ , Z) of X Σ -we will denote the faces of ∆ ω by Z 1 , . . . , Z n . We also remind the reader, that the relations between these divisors in the cohomology ring are given by the combinatorics of ∆ ω or equivalently, by that of the fan Σ. For (higher-degree) cohomology classes we will sometimes use multi-index notation, i. e. Z l expands to Z l1 1 · · · Z ln n . We will be using the weights of the torus action on the tangent bundle at fixed points of X Σ . The vertices of ∆ ω are in one-to-one correspondence with these fixed points, and we will usually denote these vertices by the greek letter σ. For any vertex σ, there are exactly d edges e 1 , . . . , e d in ∆ ω that meet at σ. Each edge of the polytope ∆ ω correspondents to T N -invariant CP 1 in X Σ . Then the tangent space T σ X Σ at σ T N -invariantly splits into the tangent lines along these subvarieties. If we denote by σ 1 , . . . , σ d the vertices that are connected by the edges e 1 , . . . , e d to σ, we will denote by ω σ σi the weight of the T N -action on T σ X Σ into the direction of e i .
When referring to a degree-2 homology class λ ∈ H 2 (X Σ , Z), we will usually give its intersection vector (λ i ) i=1,...n with the divisor classes Z i , that is λ i := Z i , λ . Note however, that the λ i have to satisfy certain linear relations to represent a degree-2 homology class. In fact we have that dim Remember that an element of M 0,m (X Σ , A) is (up to isomorphisms) a tuple (C; x 1 , . . . , x m ; f ) where C is an algebraic curve of genus zero with singularities at most ordinary double points, x i ∈ C smooth are marked points, and f : C −→ X Σ is the map to the variety X Σ . The T N -action on X Σ then induces an action of is the composition of f with the diffeomorphism ϕ t given by the action of t on X Σ .
It is then easy to see (cf. [Spi99a]), that the image of a fixed point (C; x; f ) ∈ M 0,m (X Σ , A) must be left invariant by the T N -action, or in other words, it has to live on the 1-skeleton of ∆ ω . Moreover, the marked points x i of such a stable map will have to be mapped to fixed points σ i in X Σ . The fixed-point components of M 0,m (X Σ , A) can then be characterized by so-called M 0,m (X Σ , A)-graphs Γ (see [Spi99a, Definition 6.4])-these are graphs on the 1-skeleton on ∆ ω , without loops, with decorations representing the position of the marked points and the multiplicities of the map to X Σ on the irreducible components. If Γ is such a graph, we will usually denote by M Γ the product of Deligne-Mumford spaces corresponding to the graph Γ, which is up to a finite automorphism group isomorphic to the fixed-point component in M 0,m (X Σ , A) corresponding to Γ.

Localization for the genus-0 GW invariants of toric varieties
In the setup described in the previous section, Graber and Pandharipande's virtual localization formula applies: if V is an equivariant vector bundle on the where N Γ is the so-called virtual normal bundle to M Γ and e TN denotes the equivariant Euler class. Since each ev * i (Z j ) is equal to the (standard) Euler class of an equivariant line bundle over M 0,m (X Σ , A), we obtain where ω σ(j) k is the following weight: By a careful analysis of the virtual normal bundle (see [Spi99a,Theorem 7.2]) we can compute its equivariant Euler class -before we will give its formula here, let us fix some notation. For an edge e ∈ Edge(Γ) define In this formula, we use the following notation: The edge e connects the two fixed points σ and σ d with multiplicity h. The indices i j andî j are chosen pair-wise . The homology class of the edge e is given by λ = (λ 1 , . . . , λ n ), in particular Furthermore let Vert t,s (Γ) be the set of vertices v in the graph Γ with t outgoing edges and s marked points. For such a vertex v ∈ Vert t,s (Γ) we define val(v) := t, deg(v) := s + t, and the class Here the e F are Euler classes of universal cotangent lines to marked points of M Γ = v∈Vert(Γ) M 0,deg(v) , and the indices (from 1 through t, for val(v) = t) refer to the different edges leaving the vertex v.
Proposition 1 ([Spi99a]). With this notation, the inverse of the equivariant Euler class of the virtual normal bundle has the following expression: Proposition 2 ([Spi99b, Spi00]). The integral over M Γ of the invers of the equivariant Euler class of the virtual normal bundle equals In particular, we see that if we want to generalize formula (4) to non-trivial classes β ∈ H * (M 0,m , Q), the localization formula (3) tells us that it suffices to compute the equivariant Euler classes of the restrictions of the equivariant bundles on M 0,m (X Σ , A) representing the class β, combine this class with the equivariant Euler class of the virtual normal bundle, and integrate over M Γ .

Cotangent line bundles and their restrictions to the fixed point components
In this section we will study how pull-backs of certain classes β ∈ H * (M 0,m ) localize to fixed-point components M Γ .
Let C 0,m −→ M 0,m be the universal curve, and let x i : M 0,m −→ C 0,m be the marked point sections (i = 1, . . . , m). We will denote by L i −→ M 0,m the i th universal cotangent line, that is the pull-back by x i of the relative cotangent bundle K C0,m/M0,m : For simplicity, we will restrict ourselves here to maximal sums of the line bundles L i , i.e. to those of which the rank is equal to dim M 0,m = m − 3. By Kontsevich's theorem ( [Kon92]), we know that in this case that is the Euler class of this bundle is Poincaré dual to (m−3)!/(d 1 ! · · · d m !) points; it is exactly this kind of classes β we need in order to compute quantum products of more than two factors (see equation (2)). Note that the d i fulfill the equation Proof. Since the T N -action on M 0,m (X Σ , A) is induced from the action on the image of the curve in X Σ (and which is discarded by the map π), this is obvious. Remark 5. Corollary 4 implies in particular, that the equivariant and the nonequivariant Euler classes of such pull-back bundles coincide; we will therefore use them interchangeably in these cases. Proof. If π(M Γ ) = M 0,m then the codimension of π(M Γ ) ⊂ M 0,m is at least one. Therefore e (π * E| MΓ ) = π * e E| π(MΓ) = π * (0) = 0, since rk E > dim π(M Γ ).
The Lemma implies that if e (π * E| MΓ ) = 0 for a bundle E with rk E = m − 3, the graph Γ contains only one vertex v Γ that corresponds to a stable component under the projection π to M 0,m . In other words, if we fix v Γ as root of the graph Γ, all its branches contain at most one marked point. We will call such graphs Γ simple.
Theorem 7. Let Γ be a simple M 0,m (X Σ , A)-graph, and let v Γ be the unique stable vertex of Γ andm = deg(v Γ ) be its degree. We will choose the indices of the marked pointsx 1 , . . . ,xm of M 0,m such thatx 1 , . . . ,x m are mapped by π to the marked points  We will prove the statement for the map π m+1 -the Theorem then follows by induction and Kontsevich's Theorem (equation (5)). So we want to show that (in the case d 1 + · · · + d m = m − 3) It is well-known (see e.g. [HM98]) that where the M 0,3 -bubble contains the marked points x i and x m+1 . Now note that L m+1 | Di is constant for any i = 1, . . . , m, hence e(L m+1 ) · D i = 0. This yields equation (7).

The formula for the Gromov-Witten invariants
The next Corollary summarizes what we have shown so far: .
We will now compute the integral over the fixed-point components to obtain an explicit formula for these Gromov-Witten invariants.
Theorem 9. Let Γ be a simple M 0,m (X Σ , A)-graph, and let v Γ be the unique stable vertex of Γ. Let r : Vert(Γ) −→ N be the map defined by As before let d i be non-negative integers such that d 1 + · · · + d m = m − 3. If we let E = L ⊕d1 1 ⊕ · · · ⊕ L ⊕dm m , then the following formula holds: Proof. By Kontsevich's Formula (5), it suffices to consider d 1 = m − 3, d 2 = . . . = d m = 0. In this case, by Theorem 7, the left hand side of (8) equals We therefore have to prove that MΓ e L ⊕m−3 .
For the case that v ∈ Vert t,s (Γ) is different from v Γ , we have shown in the proof of Proposition 2 (see [Spi00]) that . Hence we only have to consider the case when v = v Γ ; since it is very similar to the previous case, we will only outline its proof. As in [Spi00], let P n (x 1 , . . . , x k ) = id i=n j xd j j . Let t := val(v Γ ),m := deg(v Γ ) and r := m−3. We will also write F j instead of F j (v Γ ). Note that for the vertex v Γ , we always havem − r − 3 ≥ 0. Therefore  where R * i are the relations R i , but evaluated with respect to the quantum product instead of the cup product.
Let us consider toric manifolds of the form P P 1 (O ⊕(r−2) ⊕ O(1) ⊕ O(1)). In [CMR01], Costa and Miró-Roig have studied the three-point Gromov-Witten invariants of these manifolds and announced that they will derive the quantum cohomology ring of these manifolds in an upcoming paper. We have chosen the same example to illustrate how the formula derived in this note can make computations much easier.
We will recall some properties of P P 1 (O ⊕(r−2) ⊕ O(1) ⊕ O(1)) (for more details see [CMR01]). Its cohomology ring is given by where the relations are given by To see this, consider the fan whose one-dimensional cones are (with respect to some basis e 1 , . . . , e r in the lattice Z r ) v 1 = e 1 , v 2 = −e 1 + e r−1 + e r , v 3 = e 2 , . . . , v r+1 = e r and v r+2 = −e 2 − · · · − e r , and whose set of primitive collections is given by In [CMR01, Proposition 3.6], Costa and Miró-Roig obtain (in what follows we will freely use their notation) Hence, we will only have to compute the quantum product Z 3 ⋆ · · · ⋆ Z r+2 to get a presentation of the quantum cohomology ring. To do so, we will have to compute the Gromov-Witten invariants of the form Proof. Suppose that for given a, b and γ, the invariant (10) is non-zero. Since c 1 (X Σ ), aλ 1 + bλ 2 = 2rb, we must have 2r + deg γ = 2rb + 2r. However, the real dimension of X Σ is equal to 2r, hence 0 ≤ deg γ ≤ 2r, and therefore b ≤ 1. Now suppose b is zero and therefore deg γ = 0 as well, that is γ is a multiple of the trivial class 1 ∈ H 0 (X Σ ). If a = 0 as well, then the invariant is zero since it just equals the cup product of the cohomology classes Z 3 , . . . , Z r+2 . So suppose a > 1. Then where the γ ji run over a basis of H * (X Σ , Z). Since a > 0, at least one of a i has to be positive. On the other hand, we have Z 3 , λ 1 = . . . = Z r−1 , λ 1 = Z r+2 , λ 1 = 0.
So as soon as a i > 0 the corresponding three-point invariant in the sum above is zero. This proves the lemma.
Lemma 11. If a > 0, then all invariants of the form (10) are zero.
Proof. Suppose that for given a, b and γ, the invariant (10) is non-zero. Then, by the previous Lemma, b = 1. Hence we have to consider homology classes aλ 1 + λ 2 , and γ being a multiple of the class of top degree, say Z 1 Z 3 · · · Z r Z r+2 . The 1-skeleton of the moment polytope of X Σ has the following edges: 1. for 3 ≤ t < s ≤ r + 2, edges between σ 1,t and σ 1,s , and between σ 2,t and σ 2,s , all having homology class λ 2 ; 2. for 3 ≤ t ≤ r − 1 or t = r + 2, an edge between σ 1,t and σ 2,t of homology class λ 1 ; 3. for t = r, r + 1, an edge between between σ 1,t and σ 2,t of homology class λ 1 + λ 2 . The reader will now easily convince himself that there is no simple graph Γ in this class such that Z 3 , . . . , Z r−1 , Z r+1 , Z r+1 , Z r+2 and Z 1 Z 3 · · · Z r , Z r+2 all have non-zero equivariant Euler class on M Γ , unless a = 0. Finally note that Z r = Z r+1 as cohomology classes, which finishes the proof.
Hence we obtain and by comparing the coefficients of the q i 1 , the relation follows.
Remark 14. Note that although we derive the same presentation for the quantum cohomology ring as the one stated by Batyrev in [Bat93], the Gromov-Witten invariants that enter as structure constants into the computation of the quantum products are different from those numbers considered by Batyrev.